Product of Two Numbers, Difference of their Squares
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Another one from YouTube. I'm not going to watch it.
Given that

  x²-y²=24
  xy = 35

Find
  x + y

Solution
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Square the first equation, and get

  x⁴-2x²y²+y⁴=576  (1)

Also square the second one:

  x²y²=1225  (2)

Add 4 times quation (2) to (1):

  x⁴+2x²y²+y⁴=5476

or

   x⁴+2x²y²+y⁴=74²

Take square roots:

  x²+y²=74

Add twice the second equation xy=35

  x²+2xy+y²=144

Square root again:

  x+y=±12